QCE Mathematical Methods - Unit 3 - Differentiation of trigonometric functions and differentiation rules

Differentiation Rules | QCE Mathematical Methods

Learn when to use the chain, product and quotient rules in QCE Mathematical Methods, with worked examples and exam-style traps.

Updated 2026-05-18 - 4 min read

The chain, product and quotient rules are not three random formulas. They are pattern tools. Before you differentiate, pause and ask what kind of structure you are looking at.

Chain rule

Use the chain rule when one function is inside another function:

$ \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} $

For example, $y=\ln(3x^2+1)$ has a logarithm outside and $3x^2+1$ inside:

$ \frac{dy}{dx}=\frac{6x}{3x^2+1} $

The chain rule can have more than two layers. For:

$ y=\ln\left(\sqrt{\cos x}\right), $

there is a cosine function inside a square root inside a logarithm. One way to read it is:

$ y=\ln u,\quad u=\sqrt v,\quad v=\cos x. $

Then:

$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}. $

This gives:

$ \frac{dy}{dx} =\frac1{\sqrt{\cos x}}\cdot\frac{1}{2\sqrt{\cos x}}\cdot(-\sin x) =-\frac12\tan x. $

The point is not that you must introduce three variables every time. It is that every layer contributes one derivative factor.

For a chain-rule answer, keep the inside expression unchanged until the end. A reliable mental script is:

1. differentiate the outside function
2. copy the inside function exactly
3. multiply by the derivative of the inside

For example:

$ \frac{d}{dx}\sqrt{5x-2} =\frac{1}{2\sqrt{5x-2}}\cdot5 =\frac{5}{2\sqrt{5x-2}}. $

Writing the copied inside expression is what prevents mistakes like changing $\sqrt{5x-2}$ into $\sqrt5x-2$ or losing brackets.

Product rule

Use the product rule when two functions of $x$ are multiplied:

$ \frac{d}{dx}[u(x)v(x)]=u(x)v'(x)+v(x)u'(x) $

The order is flexible as long as both terms are there: one factor stays unchanged while the other is differentiated, then they swap.

A common product-rule pattern in Methods is a polynomial multiplied by an exponential, logarithm or trigonometric function. It is usually worth factorising the final expression because it makes stationary-point questions easier. For example, after differentiating $x^2e^{3x}$, the form:

$ xe^{3x}(3x+2) $

is more useful than:

$ 3x^2e^{3x}+2xe^{3x}, $

because the zero factors can be read directly.

Quotient rule

Use the quotient rule when one function is divided by another:

$ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right)=\frac{v(x)u'(x)-u(x)v'(x)}{[v(x)]^2} $

Students often remember this as "bottom times derivative of top, minus top times derivative of bottom, over bottom squared".

The quotient rule also carries a domain warning. The original denominator cannot be zero, and the derivative inherits that restriction. If:

$ y=\frac{\ln x}{x-2}, $

then the working must respect $x>0$ and $x\ne2$. In exam solutions, this matters most when solving derivative equations or interpreting a graph.

Worked example

Combining rules

Some questions need more than one rule. For:

$ y=\frac{\sin(2x)}{x^2+1} $

the overall structure is a quotient, but the numerator also needs the chain rule:

$ \frac{d}{dx}\sin(2x)=2\cos(2x) $

So:

$ y'=\frac{(x^2+1)2\cos(2x)-\sin(2x)(2x)}{(x^2+1)^2} $

For:

$ f(x)=x^2\sin(8x), $

the product rule gives:

$ f'(x)=x^2(8\cos(8x))+2x\sin(8x). $

For:

$ g(x)=\frac{x^2}{\sin(8x)}, $

the quotient rule gives:

$ g'(x)=\frac{\sin(8x)(2x)-x^2(8\cos(8x))}{\sin^2(8x)}. $

These two examples use similar ingredients, but the overall structure changes the rule. Multiplication and division are not interchangeable.

Common mistakes

Other traps:

• forgetting to differentiate the inside function
• reversing the subtraction order in the quotient rule
• leaving the answer expanded when a factorised form is much clearer
• treating $\sin(2x)$ as $2\sin(x)$

Rule-choice checklist

Use this order when a derivative looks messy:

1. Identify the outside structure: nesting, product, quotient or sum.
2. Apply the rule for that outside structure first.
3. Differentiate each piece, using chain rule inside pieces if needed.
4. Factorise common terms if it makes the answer clearer.

For example, $\frac{e^{2x}\ln x}{x+1}$ is a quotient overall. The numerator is a product, and $e^{2x}$ needs the chain rule. Starting with the quotient rule keeps the working organised.

Simplifying derivative answers

There is no single "best" form for every derivative, but Methods responses should be readable and useful for the next step. A good final form usually:

• keeps powers with positive exponents where practical
• factorises common exponential, polynomial or trigonometric terms
• leaves quotient-rule answers over one denominator unless expansion helps
• preserves domain restrictions for logarithms and denominators

If the next step is solving $f'(x)=0$, factorised form is usually strongest. If the next step is substitution into a value, either expanded or factorised form can be fine as long as it is equivalent.

Quick check

Sources

• QCAA Mathematical Methods subject page
• QCAA Mathematical Methods 2025 syllabus
• OpenStax Calculus Volume 1: Differentiation Rules
• OpenStax Calculus Volume 1: The Chain Rule